A simple integrate-and-fire system and various super-stable periodic orbits

Abstract

This paper studies super-stable periodic orbits and related phenomena in a simple switched dynamical system. The system repeats integrate-and-fire behavior between a constant threshold and a periodic base signal. The base signal consists of two components: the fundamental and higher frequency triangular signals. First, we demonstrate “chaos + chaos = order” such that the system exhibits chaos if the base signal is either of the two components whereas the system exhibits a super-stable periodic orbit if the base signal consists of both components. This phenomenon is confirmed experimentally and is explained theoretically. Second, we extract two key parameters and show that the system can exhibit a variety of super-stable periodic orbits. Applying a mapping procedure, existence regions of basic super-stable periodic orbits are calculated precisely.